Quasi–Monte Carlo Constructions

In this chapter and the following one, we discuss the use of low-discrepancy sampling to replace the pure random sampling that forms the backbone of the Monte Carlo method. Using this alternative sampling method in the context of multivariate integration is usually referred to as quasi–Monte Carlo. A low-discrepancy sample is one whose points are distributed in a way that approximates the uniform distribution as closely as possible. Unlike for random sampling, points are not required to be independent. In fact, the sample might be completely deterministic.

Any attempt to construct such samples requires a precise way of measuring their “uniformity”, so that we can compare different constructions and also make sure that we are indeed improving on random sampling. In fact, we are already familiar with the idea of measuring the uniformity of a point set from our discussion in Sect. 3.5 on theoretical tests for random number generators. Recall that there we were looking at the s-dimensional set ψs representing all possible sequences of s successive numbers that can be produced by the generator, and our goal was to make sure this set was “as uniform as possible”. We saw that sets ψs arising from MRGs had a lattice structure that could be assessed via the spectral test, whereas F2-linear generators were producing sets ψs whose uniformity could be measured via the concept of equidistribution through the resolution and t-value. As we will see later in this chapter, these uniformity measures can also be used for assessing the quality of low-discrepancy samples designed for quasi–Monte Carlo. But we will also see that many other measures can be used for that purpose.